G. F. Becker — New Mechanical Quadratures. 121 



derivatives and differences also require consideration the trans- 

 formation is somewhat complex, but if the #th difference is 

 technically "small," so that the (& + l)st difference is negli- 

 gible, then 



h k f(x™) = A (k) 



where A denotes finite difference. In the formulae v is 

 employed to indicate the difference of the derivatives at the 

 limits of the area to be integrated, x a and x n . Using a corre- 

 sponding notation for the finite differences and assuming that 

 the (&+l)st difference is inconsiderable 



h k v k = A n (k) -A (k) 



and this substitution may be made in the corrective terms of 

 the formulae.* 



Not all of these equations are wholly new. The first 

 term of (1) is only Simpson's rule in a new notation and if n 

 is limited to 2 it is also identical with Cotes's rule for n = 2. 

 Omitting the derivatives, equations (2) and (3) also coincide 

 with Cotes's rules for n = 4 and n = 6, but if in these equa- 

 tions n is taken at any multiple of 4 and 6, numbers quite 

 distinct from Cotes's result. All of the equations can be 

 expressed in the same form as Cotes's, but this mode of state- 

 ment seems undesirable because it masks the vital fact that a 

 reduction of the value of h increases the accuracy of the result. 

 Now, no one would think of getting a considerable quadrature 

 by Simpson's rule with the minimum value of n = 2, because 

 this rule with n = 10 gives a result the error of which 

 approaches a 625th of that incurred by taking n at 2, while if 

 in (6) n is taken at 24 instead of 12 the error is reduced 

 approximately to 1/4096 of its maximum value. 



So far as I know, equations (4), (5), and (6) are new, and 

 Cotes's numbers for n = 8 and J do not fit into the system of 

 quadratures here discussed. 



The derivative term in each of the six equations may exceed 

 the value of the remainder. If the difference of derivatives 

 in this term is denoted by v v , this is to be regarded as the 

 definite integral of f (a: (1 ' +1) ), which, like any other function of 

 real variables to be integrated, must preserve the same sign 

 between the limits of integration. If the sign does not change, 

 and if also (as Poisson and Jacobi showed) the (r+l)st deriva- 

 tive does not pass through a maximum between the limits, then 

 the final term of the equations exceeds in absolute value the 

 remainder of the series. In any case whatever let X be the 

 maximum value of f (x (r+l) ) between the limits, then the total 

 area represented by the definite integral, v T , must be less than 

 A (x a — x a ), and this substituted for v T in the corrective term 



* A discussion of the relations subsisting between derivatives and finite 

 differences may be found in Smithsonian Math. Tables, 1908, page xxxvi, or 

 elsewhere. 



